When I was SS on LL, I was drinking the required 4 litres of water a day. Average loss was 3lbs.
I increased my intake to 6 litres a day MINIMUM, some days getting to 8 or 9 litres. My average weigh losses went up to 5 lbs.
I do not know the science of it, but it seemed to increase my losses.
Another example, my father was trying to lose weight, but having just had a double knee replacement and a quadruple bypass (hence me losing all the weight because I had EXACTLY the same build as my father), he could not go SS. He had already altered his diet and had shifted a few pounds, but has plateaued. He had not lost an ounce for about a month. I advised him to start drinking plain water (which he literally drank zero of), and his weight started to come off again at about 1.5 to 2 lbs a week.
It may all just be coincidence, and I would never say 100% it was because of the water - but it seems like it to me. I cannot see how just drinking water could do this, but I have to believe what happenned to me, my father, and in fact my brother too.
I now still drink at least 4 litres of water a day, which is about twice what people recommend.
Steve